Theorem imaeq2 5102

Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Assertion

Ref	Expression
imaeq2	|- ( A = B -> ( C " A ) = ( C " B ) )

Proof of Theorem imaeq2

Step	Hyp	Ref	Expression
1		reseq2 5038	. . 3 |- ( A = B -> ( C |` A ) = ( C |` B ) )
2	1	rneqd 4991	. 2 |- ( A = B -> ran ( C |` A ) = ran ( C |` B ) )
3		df-ima 4767	. 2 |- ( C " A ) = ran ( C |` A )
4		df-ima 4767	. 2 |- ( C " B ) = ran ( C |` B )
5	2, 3, 4	3eqtr4g 2292	1 |- ( A = B -> ( C " A ) = ( C " B ) )

Syntax hints:   -> wi 4   = wceq 1398   ran crn 4755   |` cres 4756   "cima 4757

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-xp 4760  df-cnv 4762  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767

This theorem is referenced by:  imaeq2i  5104  imaeq2d  5106  fimadmfo  5604  ssimaex  5743  ssimaexg  5744  isoselem  5999  f1opw2  6269  supp0cosupp0fn  6480  fopwdom  7102  ssenen  7118  fiintim  7204  fidcenumlemrk  7237  fidcenumlemr  7238  sbthlem2  7241  isbth  7250  ennnfonelemp1  13241  ennnfonelemnn0  13257  ctinfomlemom  13262  ctinfom  13263  tgcn  15199  iscnp4  15209  cnpnei  15210  cnima  15211  cnconst2  15224  cnrest2  15227  cnptoprest  15230  txcnp  15262  txcnmpt  15264  metcnp3  15502